SAT Math Quiz: Equivalent Expressions
20 questions · exam conditions
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Equivalent ExpressionsQuestion 1 of 20

Which expression is equivalent to (2x−3)(x+5)(2x-3)(x+5)? Expand using FOIL (or distribution) and combine like terms carefully, paying attention to the signs.

2x2+13x−152x^2+13x-15
2x2+7x−152x^2+7x-15
2x2−7x−152x^2-7x-15
2x2+7x+152x^2+7x+15
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SAT Math Quiz

SAT Math Quiz: Equivalent Expressions

Practice Equivalent Expressions in SAT Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Equivalent Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for SAT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Which expression is equivalent to (2x−3)(x+5)(2x-3)(x+5)? Expand using FOIL (or distribution) and combine like terms carefully, paying attention to the signs.

  1. 2x2+13x−152x^2+13x-15
  2. 2x2+7x−152x^2+7x-15 (correct answer)
  3. 2x2−7x−152x^2-7x-15
  4. 2x2+7x+152x^2+7x+15
Explanation: To expand (2x−3)(x+5)(2x-3)(x+5), we use FOIL or distribution method. First terms: 2xcdotx=2x22x cdot x = 2x^2; Outer terms: 2xcdot5=10x2x cdot 5 = 10x; Inner terms: −3cdotx=−3x-3 cdot x = -3x; Last terms: −3cdot5=−15-3 cdot 5 = -15. Combining these gives 2x2+10x−3x−15=2x2+7x−152x^2+10x-3x-15 = 2x^2+7x-15. A common sign error occurs when multiplying the inner terms, forgetting that −3imesx=−3x-3 imes x = -3x, which would incorrectly lead to 2x2+13x−152x^2+13x-15. When using FOIL, carefully track the sign of each term throughout the multiplication.

Question 2

Which expression is equivalent to 3(2x−5)−4(x+1)+73(2x-5)-4(x+1)+7?

  1. 2x−122x-12 (correct answer)
  2. 2x−42x-4
  3. 10x−1210x-12
  4. 2x+122x+12
Explanation: We need to simplify 3(2x−5)−4(x+1)+73(2x-5)-4(x+1)+7 by distributing and combining like terms. First, distribute: 3(2x−5)=6x−153(2x-5) = 6x-15 and −4(x+1)=−4x−4-4(x+1) = -4x-4 (note the negative sign applies to both terms). This gives us 6x−15−4x−4+76x-15-4x-4+7. Combining like terms: xx terms give 6x−4x=2x6x-4x = 2x, and constants give −15−4+7=−12-15-4+7 = -12. Therefore, the expression simplifies to 2x−122x-12. A common error is forgetting to distribute the negative sign, which would incorrectly yield −4x+4-4x+4 instead of −4x−4-4x-4.

Question 3

Simplify the expression 5y−2(3y−4)+75y-2(3y-4)+7. Which expression is the simplified result?

  1. −y+15-y+15 (correct answer)
  2. −y+7-y+7
  3. 11y−111y-1
  4. −y−1-y-1
Explanation: This question asks to simplify (5y - 2(3y - 4) + 7) by distributing and combining like terms. Begin by distributing the -2: (-2 cdot 3y = -6y) and (-2 cdot (-4) = 8), so the expression becomes (5y - 6y + 8 + 7). Now combine like terms: (5y - 6y = -y) for the y-terms, and (8 + 7 = 15) for the constants, resulting in (-y + 15). A common error is forgetting the positive 8 from distributing the negative to -4, leading to incorrect constants like +7 or -1. Another mistake involves not distributing the negative sign properly, such as treating it as subtraction without flipping the signs inside. When dealing with negative distributions, double-check the signs by expanding step-by-step to ensure accuracy.

Question 4

The expression 4x2−254x^2-25 can be rewritten as a product of two binomials. Which expression is equivalent to 4x2−254x^2-25?

  1. (4x−5)(x+5)(4x-5)(x+5)
  2. (2x−5)(2x+5)(2x-5)(2x+5) (correct answer)
  3. (4x−25)(x+1)(4x-25)(x+1)
  4. (2x−5)2(2x-5)^2
Explanation: We need to factor 4x2−254x^2-25, which is a difference of squares since 4x2=(2x)24x^2 = (2x)^2 and 25=5225 = 5^2. The difference of squares pattern is a2−b2=(a−b)(a+b)a^2-b^2 = (a-b)(a+b), so 4x2−25=(2x)2−52=(2x−5)(2x+5)4x^2-25 = (2x)^2-5^2 = (2x-5)(2x+5). A common mistake is factoring out only a numerical GCF or trying to write it as (4x−5)(x+5)(4x-5)(x+5), which when expanded gives 4x2+20x−5x−25=4x2+15x−254x^2+20x-5x-25 = 4x^2+15x-25, not our original expression. Always check your factorization by expanding: (2x−5)(2x+5)=4x2+10x−10x−25=4x2−25(2x-5)(2x+5) = 4x^2+10x-10x-25 = 4x^2-25 ✓.

Question 5

Which expression is equivalent to (x+5)(x−2)(x+5)(x-2)?

  1. x2+3x+10x^2+3x+10
  2. x2−3x−10x^2-3x-10
  3. x2+7x−10x^2+7x-10
  4. x2+3x−10x^2+3x-10 (correct answer)
Explanation: The question requires expanding ((x + 5)(x - 2)) using FOIL to find the equivalent quadratic expression. Apply FOIL: First terms (x cdot x = x2x^2), Outer (x cdot (-2) = -2x), Inner (5 cdot x = 5x), Last (5 cdot (-2) = -10). Combine: (x^2 + (-2x + 5x) - 10 = x^2 + 3x - 10). A key error is using the wrong sign in the middle term, such as adding both outer and inner as positive to get +7x. Another common mistake is dropping the last term or flipping its sign to +10. For binomial multiplication, always list out FOIL steps explicitly to catch sign errors before combining.

Question 6

Which expression is equivalent to (3x+2)(x−5)(3x+2)(x-5)?

  1. 3x2−13x−103x^2-13x-10 (correct answer)
  2. 3x2−13x+103x^2-13x+10
  3. 3x2+13x−103x^2+13x-10
  4. 3x2−15x−103x^2-15x-10
Explanation: We need to expand (3x+2)(x−5)(3x+2)(x-5) using FOIL (First, Outer, Inner, Last). First: 3xcdotx=3x23x cdot x = 3x^2. Outer: 3xcdot(−5)=−15x3x cdot (-5) = -15x. Inner: 2cdotx=2x2 cdot x = 2x. Last: 2cdot(−5)=−102 cdot (-5) = -10. Combining: 3x2−15x+2x−10=3x2−13x−103x^2-15x+2x-10 = 3x^2-13x-10. Common mistakes include dropping the negative sign on −15x-15x (giving 3x2+15x+2x−103x^2+15x+2x-10) or miscalculating the constant term as positive 10. Always check signs carefully when multiplying.

Question 7

Which expression is equivalent to 3(2x−5)−2(x+7)3(2x-5)-2(x+7)?

  1. 4x+14x+1
  2. 4x−14x-1
  3. 8x−298x-29
  4. 4x−294x-29 (correct answer)
Explanation: This problem asks us to simplify the expression 3(2x−5)−2(x+7)3(2x-5)-2(x+7) by distributing and combining like terms. First, distribute the 3 to get 6x−156x-15, then distribute the -2 to get −2x−14-2x-14 (note that the negative sign must be distributed to both terms). This gives us 6x−15−2x−146x-15-2x-14. Combining like terms: 6x−2x=4x6x-2x = 4x and −15−14=−29-15-14 = -29, resulting in 4x−294x-29. A common error is forgetting to distribute the negative sign to the +7, which would incorrectly yield 4x−14x-1. When subtracting a grouped expression, always distribute the negative to every term inside the parentheses.

Question 8

Which of the following expressions is equivalent to 12a2b−18ab212a^2b-18ab^2 for all values of aa and bb?

  1. 6ab(2a−3b)6ab(2a-3b) (correct answer)
  2. 6ab(3b−2a)6ab(3b-2a)
  3. 18ab(2a3+b)18ab(\frac{2a}{3}+b)
  4. 6a2b2(a−3b2)6a^2b^2(a-\frac{3b}{2})
Explanation: Factor out the greatest common factor from 12a2b−18ab212a^2b-18ab^2: the GCF of the coefficients 12 and 18 is 6, and both terms share a factor of abab, so the GCF is 6ab6ab. Dividing each term by 6ab6ab gives 12a2b÷6ab=2a12a^2b\div 6ab=2a and −18ab2÷6ab=−3b-18ab^2\div 6ab=-3b, so 12a2b−18ab2=6ab(2a−3b)12a^2b-18ab^2=6ab(2a-3b). Choice B reverses the sign inside the parentheses, giving 6ab(3b−2a)=−12a2b+18ab26ab(3b-2a)=-12a^2b+18ab^2, the negative of the target. Choice C only factors out part of the GCF and introduces an incorrect sign. Choice D factors out an extra factor of abab, changing the degree of the expression entirely.

Question 9

Which of the following expressions is equivalent to x2−16x−4\dfrac{x^2-16}{x-4} for all x≠4x \neq 4?

  1. x−4x-4
  2. x+4x+4 (correct answer)
  3. (x−4)2(x-4)^2
  4. (x+4)2(x+4)^2
Explanation: Factor the numerator as a difference of squares: x2−16=(x−4)(x+4)x^2-16=(x-4)(x+4). Dividing by x−4x-4 (valid since x≠4x\neq 4) leaves x+4x+4. Choice A mistakenly keeps the factor being canceled instead of the remaining one. Choices C and D come from squaring one of the factors instead of simplifying the quotient.

Question 10

Which expression is equivalent to 4(x−2)−[3x−(x−5)]4(x-2)-[3x-(x-5)]? Simplify inside the brackets first, then subtract the entire bracketed expression.

  1. 2x−132x-13 (correct answer)
  2. 2x−32x-3
  3. 6x−136x-13
  4. 6x−36x-3
Explanation: To simplify 4(x−2)−[3x−(x−5)]4(x-2)-[3x-(x-5)], we must work from the inside out. First, simplify inside the brackets: 3x−(x−5)=3x−x+5=2x+53x-(x-5) = 3x-x+5 = 2x+5. Now we have 4(x−2)−[2x+5]4(x-2)-[2x+5]. Distributing the 4 gives 4x−84x-8, and subtracting the bracketed expression gives 4x−8−2x−5=2x−134x-8-2x-5 = 2x-13. The key error to avoid is forgetting to distribute the negative sign when removing the brackets, which would incorrectly yield 4x−8−2x+5=2x−34x-8-2x+5 = 2x-3. When subtracting a bracketed expression, change the sign of every term inside.

Question 11

Which expression is equivalent to 2(4y−3)−(y+5)2(4y - 3) - (y + 5)?

  1. 9y−19y - 1
  2. 7y+117y + 11
  3. 8y−88y - 8
  4. 7y−117y - 11 (correct answer)
Explanation: Compute 8y−6−y−5=7y−118y - 6 - y - 5 = 7y - 11. The distractors come from not distributing the negative or miscombining constants and coefficients.

Question 12

Which of the following expressions is equivalent to (x−3)3−(x−3)2(x - 3)^3 - (x - 3)^2?

  1. (x−3)2(x−6)(x - 3)^2(x - 6)
  2. (x−3)2(x−4)(x - 3)^2(x - 4) (correct answer)
  3. (x−3)3(x−1)(x - 3)^3(x - 1)
  4. (x−3)(x−4)2(x - 3)(x - 4)^2
Explanation: Factor the common (x−3)2(x - 3)^2: (x−3)2[(x−3)−1]=(x−3)2(x−4)(x - 3)^2[(x - 3) - 1] = (x - 3)^2(x - 4). The other options reflect incorrect factoring or exponents.

Question 13

Which expression is equivalent to 2x(3x−4)−5(1−2x)2x(3x-4)-5(1-2x)? Distribute in both terms and then combine like terms.

  1. 6x2−18x−56x^2-18x-5
  2. 6x2−18x+56x^2-18x+5
  3. 6x2−8x−56x^2-8x-5
  4. 6x2+2x−56x^2+2x-5 (correct answer)
Explanation: To simplify 2x(3x−4)−5(1−2x)2x(3x-4)-5(1-2x), distribute both parts carefully. First: 2x(3x−4)=6x2−8x2x(3x-4) = 6x^2 - 8x. Second: −5(1−2x)=−5+10x-5(1-2x) = -5 + 10x (note the positive 10x10x). Combining: 6x2−8x−5+10x=6x2+2x−56x^2 - 8x - 5 + 10x = 6x^2 + 2x - 5. The critical step is correctly distributing −5-5 to get +10x+10x, since −5imes(−2x)=+10x-5 imes (-2x) = +10x. Many students incorrectly get −10x-10x, forgetting that negative times negative equals positive.

Question 14

Which of the following expressions is equivalent to 3(x−2)+2(x+5)3(x-2)+2(x+5)?

  1. 5x+45x+4 (correct answer)
  2. 5x−45x-4
  3. x+4x+4
  4. 5x+165x+16
Explanation: Distribute and combine like terms: 3x−6+2x+10=5x+43x-6+2x+10=5x+4. The other choices result from sign mistakes, failing to distribute, or incorrect constant addition.

Question 15

Factor x2−9x+20x^2 - 9x + 20.

  1. (x−5)(x−4)(x - 5)(x - 4) (correct answer)
  2. (x+5)(x−4)(x + 5)(x - 4)
  3. (x−5)(x+4)(x - 5)(x + 4)
  4. (x−10)(x+2)(x - 10)(x + 2)
Explanation: Find two numbers that multiply to 20 and add to -9: -5 and -4, giving (x−5)(x−4)(x - 5)(x - 4). The other options either give the wrong middle term or the wrong product.

Question 16

A rectangle's area is modeled by x2−9x^2-9. To factor the expression and find possible side lengths, which expression is equivalent to x2−9x^2-9?

  1. (x+3)(x−3)(x+3)(x-3) (correct answer)
  2. x(x−9)x(x-9)
  3. (x−3)2(x-3)^2
  4. (x+9)(x−1)(x+9)(x-1)
Explanation: The question requires factoring (x2x^2 - 9) to find an equivalent expression representing possible side lengths of a rectangle with that area. Recognize that (x2x^2 - 9) is a difference of squares, which factors as ((x + a)(x - a)) where (a2a^2 = 9), so (a = 3). Thus, it factors to ((x + 3)(x - 3)). Verify by expanding: (x cdot x = x2x^2), (x cdot (-3) = -3x), (3 cdot x = 3x), and (3 cdot (-3) = -9), combining to (x^2 + (-3x + 3x) - 9 = x^2 - 9). A key error is confusing it with a perfect square like ((x - 3)^2 = x^2 - 6x + 9), which adds a middle term. Another mistake is uneven factoring, such as (x(x - 9) = x^2 - 9x), missing the constant. For factoring questions, check your answer by expanding back to the original to confirm equivalence.

Question 17

If 4(2x−1)−k4(2x-1)-k is equivalent to 8x−138x-13 for all values of xx, what is the value of kk? Distribute first, then match constants to avoid sign mistakes.

  1. −9-9
  2. 99 (correct answer)
  3. 1111
  4. 1313
Explanation: The question asks for the value of (k) such that (4(2x - 1) - k) is equivalent to (8x - 13) for all (x), by matching coefficients after simplifying. Start by distributing the 4: (4 cdot 2x = 8x) and (4 cdot (-1) = -4), so the left side is (8x - 4 - k). For equivalence, the x-coefficients already match (both 8), so set the constants equal: (-4 - k = -13). Solve for k: (-k = -13 + 4 = -9), so (k = 9). A common error is mishandling signs when isolating k, such as adding 4 to both sides incorrectly to get k = 13. Another mistake is not distributing first, leading to mismatched terms like comparing 2x to 8x. When finding missing constants in equivalent expressions, expand fully and equate like terms to solve systematically.

Question 18

Which expression is equivalent to 2x2−8x2x^2-8x written in factored form?

  1. 2x(x−4)2x(x-4) (correct answer)
  2. x(2x−4)x(2x-4)
  3. 2(x2−4)2(x^2-4)
  4. 2x(x+4)2x(x+4)
Explanation: To factor 2x2−8x2x^2-8x, we first identify the greatest common factor (GCF). Both terms have a factor of 2 and a factor of xx, so the GCF is 2x2x. Factoring out 2x2x: 2x2÷2x=x2x^2÷2x=x and −8x÷2x=−4-8x÷2x=-4. Therefore, 2x2−8x=2x(x−4)2x^2-8x=2x(x-4). A common mistake is to factor out only part of the GCF, such as just 22 or just xx, giving 2(x2−4x)2(x^2-4x) or x(2x−8)x(2x-8). While these are correct, they're not completely factored. Always factor out the complete GCF in one step.

Question 19

Expand and simplify (2x−3)2(2x-3)^2. Write your final answer as a quadratic in standard form.

  1. 4x2−94x^2-9
  2. 4x2−12x+94x^2-12x+9 (correct answer)
  3. 4x2−6x+94x^2-6x+9
  4. 2x2−12x+92x^2-12x+9
Explanation: We need to expand (2x−3)2(2x-3)^2, which means (2x−3)(2x−3)(2x-3)(2x-3). Using the pattern (a−b)2=a2−2ab+b2(a-b)^2=a^2-2ab+b^2, we get (2x)2−2(2x)(3)+32=4x2−12x+9(2x)^2-2(2x)(3)+3^2=4x^2-12x+9. Alternatively, using FOIL: First gives 4x24x^2, Outer gives −6x-6x, Inner gives −6x-6x, and Last gives 99, so we have 4x2−6x−6x+9=4x2−12x+94x^2-6x-6x+9=4x^2-12x+9. The most common error is getting the wrong middle term by forgetting to double the product of the two terms, writing −6x-6x instead of −12x-12x. Remember that (a−b)2(a-b)^2 always has a negative middle term equal to −2ab-2ab.

Question 20

Factor the trinomial 6x2+11x+36x^2+11x+3. Choose the correct factorization. Several choices seem

  1. (3x−1)(2x−3)(3x-1)(2x-3)
  2. (6x+1)(x+3)(6x+1)(x+3)
  3. (2x+1)(3x+3)(2x+1)(3x+3)
  4. (3x+1)(2x+3)(3x+1)(2x+3) (correct answer)
Explanation: To factor 6x2+11x+36x^2+11x+3, we need two binomials whose product gives this trinomial. The first terms must multiply to 6x26x^2 (options: 6xcdotx6x cdot x or 3xcdot2x3x cdot 2x) and the last terms must multiply to 33 (options: 3cdot13 cdot 1). Testing (3x+1)(2x+3)(3x+1)(2x+3): First terms give 6x26x^2, outer gives 9x9x, inner gives 2x2x, last gives 33. Middle term: 9x+2x=11x9x+2x = 11x ✓. Therefore, 6x2+11x+3=(3x+1)(2x+3)6x^2+11x+3 = (3x+1)(2x+3). A common error is pairing factors incorrectly, like (6x+1)(x+3)(6x+1)(x+3) which gives a middle term of 19x19x. When factoring trinomials with leading coefficient ≠ 1, systematically test factor pairs.